Wyniki wyszukiwania - BazTech

Ograniczanie wyników

2 Snopek K. M.

Znaleziono wyników: 2

Liczba wyników na stronie

Wyniki wyszukiwania

Wyszukiwano:
w słowach kluczowych: hypercomplex signals

Sortuj według:

Ogranicz wyniki do:

Studies on complex and hypercomplex multidimensional analytic signals

Snopek K. M.

Prace Naukowe Politechniki Warszawskiej. Elektronika

2013

z. 190

1--169

Within the last twenty years, the theory of complex and hypercomplex multidimensional signals has been still developing and finding new interesting applications in various fields This monograph, devoted to analytic complex and hypercomplex signals, presents mutual relations between these two approaches in signal- and frequency domains. Complex and hypercomplex signals are n-dimerisional (n-D) generalizations of corresponding complex and hypercomplex numbers belonging to Cayley-Dickson and Clifford algebras. The brief look into the properties of the Cayley-Dickson algebras of quaternions, octonions and sedenions and also into the Clifford algebras of biquaternions and bioctonions is presented. The tables of multiplication of imaginary units in all considered algebras have been constructed. The main part of the book concerns complex and hypercomplex multidimensional signals with spectra defined using various Fourier transformations (FTs). The n-D Clifford FT is recalled and the new n-D Cayley-Dickson FT is introduced. For n = 2, both are equivalent and known as the Quaternion FT (QFT). For n = 3, the new Cayley-Dickson FT, named the Octonion FT (OFT), is introduced The relation between the 3-D complex FT and the OFT is derived basing on multiplication rules in the algebra of octonions. The n-D complex and hypercomplex analytic signals are defined in the signal domain using the n-D Convolution with the corresponding n-D delta distribution. Such a idea was proposed by Hahn in 1992 for multidimensional complex signals with single-orthant spectra. Here, it is applied for quaternion and octonion analytic signals. Their spectra are defined in analogy to the Hahn's approach, as the inverse QFT (resp. inverse OFT) of a single-quadrant (resp. single-octant) spectrum. The complete set of definitions of 2-D and 3-D analytic signals with spectra in all orthants is included. It has been noticed that there are some conjugate pairs between them and basing on this fact, the lower rank signals have been defined. Basing on definitions of complex, quaternion and octonion analytic signals, it is possible to define the local amplitudes and local phase functions of a signal. As the dimension of a signal is higher, the number of polar components increases. It has been shown that in 2-D, a signal is completely defined by two amplitudes and two phase functions in the complex case and in the quaternion case, by a single amplitude and three phases represented by Euler angles. In case of separable signals, the number of components is reduced. In 3-D, the complex analytic signal is represented by four amplitudes and four phase functions, while the octonion analytic signal has a single amplitude and seven phase functions. In this work, the hypothesis concerning the mutual relations between the complex and octonion polar representation is discussed.

New Hypercomplex Analytic Signals and Fourier Transforms in Cayley-Dickson Algebras

Snopek K. M.

Electronics and Telecommunications Quarterly

2009

Vol. 55, No 3

403-415

In the paper, the definitions of the new hypercomplex Fourier transform and the new hypercomplex analytic signal are recalled. They have been derived basing on Cayley-Dickson algebra multiplication rules. Two other definitions of multidimensional Fourier transforms are presented, i.e., the classical one and the Clifford FT. The relation between them for 3-D real signals is derived. The differences between three complex/hypercomplex FTs are shown regarding the even-odd spectrum components in the 3-D case and norms/pseudonorms of 3-D analytic signals.